Some local and global properties of secant varieties of nonsingular projective curves

TL;DR

This paper investigates local and global geometric properties of secant varieties of algebraic curves, providing new descriptions, cohomology computations, and recursive formulas that extend previous research.

Contribution

It offers a detailed description of tangent cones, computes cohomology groups of secant sheaves, and introduces a recursive formula for Hilbert polynomials, advancing understanding of secant varieties.

Findings

Description of tangent cones of secant varieties

Cohomology groups of secant sheaves computed

Recursive formula for Hilbert polynomials derived

Abstract

The main goal of this paper is to study some local and global properties of secant varieties of algebraic curves. These results complement our previous work [8] by addressing issues given therein and providing solutions to problems raised subsequently. Specifically, we show a description of tangent cones of secant varieties of curves, and compute the cohomology groups of secant sheaves on symmetric products of curves, which answers a question posed in [8] and leads to a recursive formula for Hilbert polynomials of secant varieties of curves. In the appendix, we present a cohomological approach to arithmetical Cohen--Macaulayness of secant varieties of curves, completing the proof in [8].